Coherent states in mathematical physics

Coherent states have been introduced in a physical context, first as quasi-classical states in quantum mechanics, then as the backbone of quantum optics and they are described in that spirit in the article Coherent states (see also). However, they have generated a huge variety of generalizations, which have led to a tremendous literature in mathematical physics. In this article, we sketch the main directions of research on this line. For further details, we refer to several existing surveys .

Let ${\displaystyle {\mathfrak {H}}\,}$ be a complex, separable Hilbert space, ${\displaystyle X}$ a locally compact space and ${\displaystyle d\nu }$ a measure on ${\displaystyle X}$. For each ${\displaystyle x}$ in ${\displaystyle X}$, denote ${\displaystyle |x\rangle }$ a vector in ${\displaystyle {\mathfrak {H}}}$. Assume that this set of vectors possesses the following properties:

...
Wikipedia